Assume that the continuum is $\aleph_2$ in our ground model $V$. Suppose that $(P_n \, : \, n \in \omega)$ is a fully supported iteration of length $\omega$. Suppose further that the factors of the iteration are proper and add for every step $\omega_1$-many Cohen reals over the previous model.

If we let $P_{\omega}$ be the inverse limit, then there will be new reals which were not added by one of the $P_n$'s. Will there be a real $r$ in $V^{P_{\omega}}$, such that $r$ is Cohen over every $V^{P_n}$?

I am also highly interested in the dual situation, where we replace Cohen forcing with Random forcing in the above. Can we guarantee that there is a real $r$ which is Random over all $V^{P_n}$'s? If not, is it possible to feed in countably many proper factors to the iteration, such that we can exclude reals which are Random/Cohen over the $V^{P_n}$'s

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